Rare Mutations in Evolutionary Dynamics

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by  Anna Lisa Amadori

RARE MUTATIONS IN EVOLUTIONARY DYNAMICS
ANNALISA AMADORI1, ANTONELLACALZOLARI2,
ROBERTO NATALINI3,BARBARA TORTI2
3 Abstract
1
0 In this paper we study the effect of rare mutations, driven by a marked point
2
process, on the evolutionary behavior of a population. We derive a Kolmogorov

n equation describing the expected values of the different frequencies and prove some
a
rigorous analytical results about their behavior. Finally, in a simple case of two
J
different quasispecies, we are able to prove that the rarity of mutations increases the
8
2 survival opportunity of the low fitness species.

] 2000 Mathematics Subject Classification: 92D25 (35R09, 60G55)
S
D Keywords: Coevolutionarydynamics,mutations,markedpointprocesses,partialintegro-
differential equations.
.
h
t
a 1. Introduction
m
Evolutionary dynamics describes biological systems in terms of three general prin-
[
ciples: replication, selection and mutation. Each biological type – a genome or a
2
phenotype as well as a species – is described by its reproduction rate, or fitness. In
v
0 force of selection, a population evolves and changes its fitness landscape. Genetic
7 changes can help in reaching some local optimum, or open a path to a new fitness
1
peak, but sometimes they may drift population away from a peak, especially if the
4
mutation rate is high. See Nowak (2006) for an extensive account of the state of the
.
1
art concerning evolutionary dynamics.
1
2 A mixed population of constant size constituted by a fixed number of different
1 types is characterized by the vector collecting the relative abundance of each type:
:
v x = (x0,x1,...,xd). By virtue of evolution, this vector draws a path in the simplex
i
X d
S = x = (x ,x ...x ) ∈Rd+1 : x = 1, x ≥ 0 as k = 0,1...d .
r 0 1 d k k
a ( )
k=0
X
The mechanism of replication/selection is well described by an ordinary differential
equation, where the relative fitness measures the balance between death and birth of
individuals. Denoting by f the absolute fitnessof any k-type, by f = (f ,x ,...,f )
k 0 1 d
the fitness vector, and by f¯= x·f the mean fitness of the population, this equation
reads
dx
(1.1) k = f −f¯ x , as k = 0,1,...d.
k k
dt
Several shapes have been (cid:0)propose(cid:1)d for the absolute fitness. When one is modelling
phenotypes, the choice of a constant fitness seems fair, but, starting with the seminal
work by Maynard Smith and Price (1973), an important amount of research deals
withideasarisingfrommathematical gametheory, seeHofbauer and Sigmund(1998)
1
2 A.L.AMADORI,A.CALZOLARI,R.NATALINI,B.TORTI
and references therein. In that framework, the individual fitness is taken as a linear
function of the population x, i.e. f(x) = Ax, where A is the payoff matrix that
rules the interplay between different strategists. In this case, equation (1.1) is the
celebrated replicator equation introduced by Taylor and Jonker (1978).
Concerningmutations,ithastobementionedthequasispeciesequationintroduced
by Eigen and Schuster (1979), where constant fitness were considered. This choice
modifies equation (1.1) into
d
dx
(1.2) k = f q x −f¯x , as k = 0,1,...d.
i ik i k
dt
i=0
X
Herethecoefficient q expresstheproportionofoffspringofk-typefromaprogenitor
ik
i,whichshowsupatanyprocreation. Itisclearthatq = δ givesbacktheequation
ik ik
(1.1). When the fitness vector is given by the relation f(x) = Ax, as suggested by
evolutionary game theory, then equation (1.2) is the well-known replicator-mutator
equation, also knownas selection-mutation equation, studiedin Stadler and Schuster
(1992).
As a matter of fact, mutations introduce a random ingredient into evolution, that
is not enough emphasized in (1.2). Traulsen et al. (2006) pointed out that equation
(1.2) can be recovered by assuming that the population follows a generalized Moran
process and taking the limit for large population size. Champagnat et al. (2008)
and Jourdain et al. (2012) showed that various macroscopic diffusion models can
be derived by the same individual stochastic process, by performing different types
of rescaling. We also mention Dieckmann and Law (1996), where a macroscopic
dynamics is deduced by an individual based stochastic description of the mutation
process.
Here,weprefertotakeamoremacroscopicviewpoint,whichhowevertakesstrongly
into account the different regime of mutation processes. We start by modelling the
stochastic dynamics at the level of the frequency vector x. In order to capture the
“rarity” of mutations, we assume that they are driven by some point processes. As a
result, the random path of the frequency vector x in the simplex is not continuous.
This happens because mutations arise at a different time-scale with respect to repli-
cation and selection, and this assumption constitutes the main novelty of the present
paper.
The stochastic dynamics for frequencies is introduced and studied in Section 2. In
Section3,aKolmogorov equationdescribingtheexpectedvaluesisrigorouslyderived
andstudiedinitsanalytical aspects. Because ofthepointprocess,suchequationis of
integro-differential type. Global existence of mild solutions to this equation is proved
as well as some usefulbasic estimates. Next, in Section 4, we deal with theparticular
case of two different quasispecies. We prove the existence of a stable equilibrium,
that in general is greater or equal to the one of the standard quasispecies equation,
to point out the fact that the rarity of mutations increases the survival opportunity
of the low fitness species. In this sense, the single relevant contribution of our paper
is to show that the equilibrium position depends not only on the global amounts of
mutations, but also on the time intensity of the process driving mutations. As the
time intensity goes to infinity, the standard quasispecies equilibrium is recovered.
But when mutation are concentrated in few, very rare events, a different equilibrium
arises. We prove this occurrence in a rigorous way in Section 4, using some refined
RARE MUTATIONS IN EVOLUTIONARY DYNAMICS 3
uniformestimates of thederivatives of thesolution, inthesimplecaseof twodifferent
quasispecies, in a given range of the parameters of the model.
2. The stochastic model
We introduce here a stochastic differential equation that describe the population
distribution, under rare sudden mutations. Let us first revise the selection-mutation
equation (1.2). To underlinetheeffect of mutation, we follow Hofbauer and Sigmund
(1998) and introduce the “effective mutation matrix” M = Q−I. If no mutation
occurs, M is the null matrix; in general M = (m ) with m = q ∈ [0,1]
ik i,k=0,1,...,d ik ik
if i 6= k and m = q −1 = − m ∈ [−1,0]. Equation (1.2) can be written as
kk kk ki
i6=k
P
d
d
(2.1) x = f (x)−f¯(x) x + f (x)m x ,
k k k i ik i
dt
i=0
(cid:0) (cid:1) X
for t > 0 as k = 0,1,...d. Here and henceforth we have assumed that all f (x)
k
are nonnegative. We remark that the term f (x)−f¯(x) x stands for the homoge-
k k
neous reproduction steered by the replicator equation, the term f (x)m x ≥ 0
i ik i
(cid:0) (cid:1)
i6=k
describestheincreasingofthefrequencyofk-typeyieldedbybirthPofk-individualsby
mutation, and the term f (x)m x ≤ 0 measures the decreasing of the frequency of
k kk k
k individuals caused by the birth of mutated descendants by k-type progenitors. The
underlying assumption is that mutations happen at the same time-scale as homoge-
neous reproduction: at any procreation, a fixed proportion of the progenies shows
up a mutant trait. It seems however more realistic to describe mutations as sudden
changes in the population distribution: we thus assume here that they are driven
by some marked point processes. To be more precise, let (T ,Z ) be a sequence
n n n
of random times and random marks on a filtered probability space (Ω,F,(F ),P).
t
The marks are chosen in the mark space Z = {(i,k) ∈ {0,...,d}2, i 6= k}, i.e. it
is assumed that each mutation has progenitor of a fixed type and descendants of a
different unique type. Set
N = I , Nik = I I , i 6= k, t ≥ 0,
t (Tn≤t) t (Zn=(i,k)) (Tn≤t)
n n
X X
which definethe processes that count respectively the total number of mutations and
the number of mutations with ancestor i and descendants k, so that N = Nik.
t i6=k t
The intensity of Nik should depend on the “genetic distance” from type i to type k.
t P
Besides, it is affected also by selection: the larger the fitness fi(xt−), the more type
i reproduces, the more often its offspring shall suffer a mutation. Hence we take as
(F )-intensity kernel
t
λt(dz) = λikfi(xt−)I(i,k)(dz),
i6=k
X
whereλ arepositive constants. Itfollows that, foreach i6= k, Nik isapointprocess
ik
with (Ft)-intensity equal to λikfi(xt−). We remark that when the fitness vector f is
constant then, for each i 6= k, Nik is a classical Poisson process of parameter λ fi
ik
(see also Remark 2.1).
The proportion of the offspring of i-individuals showing a k-type by effect of mu-
tation is taken constant and denoted by γ ∈ (0,1]. A stochastic dynamics for the
ik
4 A.L.AMADORI,A.CALZOLARI,R.NATALINI,B.TORTI
relative frequencies arises
(2.2) dxk = f (x )−f¯(x ) xkdt+ γ xidNik − γ xkdNki,
t k t t t ik t t ki t t
i6=k i6=k
(cid:0) (cid:1) X X
as k = 0,1...d. We next address to well-posedness of the S.D.E. (2.2). In view
of writing its infinitesimal generator (and, later on, its Kolmogorov equation), we
introduce the vector valued function a(x) = (a (x),a (x)...a (x)), with
0 1 d
(2.3) a (x) =− f (x)−f¯(x) x , k = 0,1...d,
k k k
and the first order discrete(cid:0)non-local fun(cid:1)ctional
(2.4) Jφ(x) = λ f (x)[φ(x+γ x (e −e ))−φ(x)].
ij i ij i j i
i6=j
X
Here e stands for the unit vector pointing in the direction of the jth axis. Taking
j
into account that for (i,k) 6= (j,l) the processes Nik and Njl have no common jump
times, Ito’s formula for semimartingales gives
(2.5) dφ(xt) = (−a(xt)·φ(xt)+Jφ(xt−))dt+dMt
for each function φ∈ C1(Rd+1). Here M is the (F )-martingale defined by
b t
dMt = φ(xt− +γikxit−(ek −ei))−φ(xt−) dMtik,
i6=k
X(cid:2) (cid:3)
t
with Mik = Nik − λ f (x )ds. The S.D.E. equation (2.2), endowed with any
t t ik i s
Z0
random initial position F -measurable, is well-posed in the weak sense stated by
0
nextlemma. Moreover itisconsistentwithfrequencymodeling,i.e.whenthestarting
position is in S then the solution stays in S for all t ≥ 0, a.s.. Actually, the next
Lemma holds.
Lemma 2.1. Let f : Rd+1 → Rd+1 be a Lipschitz continuous function with compact
support containing the simplex S and such that |f(x)| =6 0 if x ∈ S. Let B be the
operator defined by
Bφ(x) = −a(x)·∇φ(x)+I (x)Jφ(x).
S
Then for every probability measure π assigned on (Rd+1,B(Rd+1)), the martingale
0
problem for (B,π ) is well-posed, that is there exists a filtered probability space and a
0
(d+1)-dimensional Markov process (X , t ≥ 0) with initial distribution π on it such
t 0
that
t
φ(X )−φ(X )− Bφ(X )ds
t 0 s
Z0
is a martingale. This process is unique in law, that is if (X˜ , t ≥ 0) is a different
t
solution of the martingale problem for (B,π ), then X and X˜ have the same finite
0
dimensional distributions.
Moreover, if the support of π is contained in S then for all t ≥ 0, X ∈ S a.s..
0 t
Proof. For all φ∈ C1(Rd+1), I (x)Jφ(x) can be written as
b S
λ(x) φ x+I (x)y −φ(x) m (dy)
S x
Rd+1
Z h i
(cid:0) (cid:1)
RARE MUTATIONS IN EVOLUTIONARY DYNAMICS 5
where
λ(x) = λ f (x)
ik i
i6=k
X
and m (dy) is the probability measure on Rd+1,B(Rd+1) defined by
x
λikfi(x) (cid:0) (cid:1)
δ (dy), if x ∈ S
λ(x) γikxi(ek−ei)
(2.6) m (dy) =
x  i6=k
X
 any probability measure, if x 6∈ S.
Note that by the hypotheses λ(x) 6= 0 when x ∈S.

Moreover it is easy to see by using Skorohod construction for random variables
that I (x)Jφ(x) can also be expressed in the form
S
φ x+K(x,ζ) −φ(x) ν(dζ)
ZΞ
(cid:2) (cid:0) (cid:1) (cid:3)
where (Ξ,Υ) is a measurable space, (x,ζ) → K(x,ζ) is a measurable bounded func-
tion on Rd+1×Ξ with values in Rd+1 and ν(dζ) is a σ-finite measure on (Ξ,Υ).
More precisely the previous equality holds for (Ξ,Υ)= (0,1)d ×R+,B((0,1)d)⊗
B(R+) , with general element denoted by ζ = (u ,u ,...u ,θ),
0 1 (cid:0)d−1
(cid:1) ν(dζ) = du0du1...dud−1dθ,
and the function K constructed as follows (see Calzolari and Nappo (1996)). Fixed
x ∈ S, let Y = (Y ,...,Y ) be a random vector with law m (dy) defined by (2.6).
0 d x
Let y → F (y ) be the distribution function of Y and moreover, for n = 1,...,d,
0 0 0 0
let F (y |y ,...,y ) be the distribution function of Y given Y = y ,...,Y =
n n 0 n−1 n 0 0 n−1
y , so that
n−1
m (dy) = F (dy )F (dy |y )...F (dy | y ,...,y ).
x 0 0 1 1 0 d d 0 d−1
Finally denote by F−1 the generalized inverse of F , for n = 0,1,...,d. Then for
n n
x ∈ S let K(x,·) :Ξ → Rd+1 be defined by
K (x,ζ) = F−1(u ),
0 0 0
K (x,ζ) = F−1 u |K (x,ζ) ,
1 1 1 0
Kn(x,ζ)= Fn−1(cid:0)un |K0(x,ζ)(cid:1),K1(x,ζ),...,Kn−1(x,ζ) , n =1,...,d−1,
and (cid:0) (cid:1)
K (x,ζ)= I (θ)F−1 θ/λ(x) |K (x,ζ),K (x,ζ),...,K (x,ζ) .
d (0,λ(x)) d 0 1 d−1
(cid:16) (cid:17)
For x ∈/ S let K(x,·) be identically zero.
It is to remark that the above construction implies that, for x ∈ S, we have
(2.7) ν ζ ∈ Ξ,|K(x,ζ)| =6 0 = λ(x).
In fact the distribution func(cid:0)tions used in the con(cid:1)struction have no jumps at zero so
that none of the general inverses yields zero on a set of positive measure. Moreover
for all x and ζ, |K(x,ζ)| ≤ 2. Then existence of a solution of the martingale problem
for (B,π ) follows by existence of a solution of the SDE
0
t t
(2.8) Xt = X0− a(Xs)ds+ K(Xs−,ζ)N(ds×dζ), t ≥ 0
Z0 Z0 ZΞ
6 A.L.AMADORI,A.CALZOLARI,R.NATALINI,B.TORTI
where N(dt×dζ) is a Poisson random measure on Rd+1,B(Rd+1) with mean mea-
sure dt×ν(dζ) and X is a random variable on the same probability space with dis-
0
(cid:0) (cid:1)
tribution π . A strong non-explosive Markov solution of (2.8) exists by Athreya et al.
0
(1988), wheremoregeneralSDEwithjumpsaretreated.Indeed,underourregularity
assumptiononf,notonlythedriftcoefficientverifiessub-lineargrowthandLipschitz
condition but also the intensity of the point process which counts the total number
of jumps in (2.8) is bounded. In fact (2.7) joint with the regularity of f gives
supν ζ ∈ Ξ,|K(x,ζ)| =6 0 = supλ(x) <+∞.
x∈S x∈S
(cid:0) (cid:1)
Again we refer to Athreya et al. (1988) for deriving uniqueness in law of (X , t ≥ 0).
t
Finally, following Athreya et al. (1988), the construction of the strong solution
of equation (2.8) uses sequentially the deterministic and the stochastic part of the
dynamic. Soifthesupportofπ iscontainedinS theneverytrajectoryofthesolution
0
verifies X ∈ S, for all t ≥ 0, when either the deterministic dynamic or the stochastic
t
dynamic do not allow to the trajectories starting in S to leave S. As it is well-
known this is true for the deterministic dynamic. As far as the stochastic dynamic
is concerned, it is sufficient to note that when x ∈ S then x+γ x (e −e )∈ S. (cid:3)
ik i k i
Proposition 2.1. The frequencies process is well-defined in law for all t ≥ 0.
Proof. It follows immediately by previous lemma recalling (2.5), i.e. that the fre-
quencies process starting at x ∈ S solves for all t ≥ 0 the martingale problem for
(B,δ ). (cid:3)
x
Remark 2.1. Let us remark that when the fitness vector f assumes constant value
on S, then the point processes Nik are classical Poisson processes of parameter λ fi
t ik
and (2.2) can be written in form (2.8) with
Ξ = Z = {(i,k) ∈{0,...,d}2, i6= k}, K(x,(i,k)) = I (x)γ x (e −e )
S ik i k i
N([0,t]×(i,k)) = Nik, ν({(i,k)}) = λ fi.
t ik
So Athreya et al. (1988) directly applies to our model, and the frequencies process is
the strong solution of equation (2.2) so that
d
dxk = f (x )−f¯(x ) xk + f (x )λ γ xi dt
t k t t t i t ik ik t
" #
i=0
(cid:0) (cid:1) X
+ γ xi dMik − γ xk dMki.
ik t− t ki t− t
i6=k i6=k
X X
By taking m = λ γ as i 6= k, m = − λ γ , the stochastic dynamics (2.2)
ik ik ik kk ki ki
i6=k
is nothing but the quasispecies equation (2.1)P, perturbed by a martingale term.
3. The Kolmogorov equation
We next address to the expected value of the frequencies process
u (x,t) = E xk x = x ,
k t 0
as k = 0,1,...d, and deduce rigorously th(cid:16)at i(cid:12)t satisfi(cid:17)es its Kolmogorov equation.
(cid:12)
RARE MUTATIONS IN EVOLUTIONARY DYNAMICS 7
Proposition 3.1. For each k = 0,1...d we have
∂ u (x,t)+a(x)·∇u (x,t) = Ju (x,t), x ∈ S, t > 0
t k k k
u (x,0) = x , x ∈ S, t = 0,
k k
(cid:26)
We recall that a and J have been introduced in (2.3) and (2.4), respectively.
Proposition 3.1 follows readily by next result about the semigroup on B(RD+1)
defined by
T φ(x) = E(φ(Xx)).
t t
Here, Xx denotes the solution at time t of (2.2), with deterministic starting position
t
x ∈ Rd+1.
Lemma 3.1. Let f and B be as in Lemma 2.1. Then B is the infinitesimal generator
of (T , t ≥ 0) with domain D(B) containing C2 (RD+1). Moreover for each φ ∈
t K
C2 (RD+1) the scalar function (x,t) → u (x,t) = T φ(x) satisfies the Kolmogorov
K φ t
equation
∂ u (x,t)+a(x)∇u (x,t) = Ju (x,t),
t φ φ φ
u (x,0) = φ(x).
φ
(cid:26)
Proof. Any function φ ∈C2 belongs to the domain of B since
K
T φ(x)−φ(x)
t
lim sup −Bφ(x) = 0.
t→0+x∈RD+1 t
(cid:12) (cid:12)
(cid:12) (cid:12)
In fact Ito’s formula joint with Fu(cid:12)bini’s theorem gives (cid:12)
t
(3.1) T φ(x) = φ(x)+ T Bφ(x)ds
t s
Z0
so that the above limit coincides with
t
(3.2) lim t−1 sup E(Bφ(Xx)−Bφ(x))ds = 0.
s
t→0+ x∈RD+1(cid:12)Z0 (cid:12)
(cid:12) (cid:12)
Then equality (3.2) follows by co(cid:12)nsidering that for all x ∈ Rd(cid:12)+1 the regularity of f
and φ implies that
E |Bφ(Xx)−Bφ(x)| ≤ C(φ)s,
s
with C(φ) a positive consta(cid:0)nt depending on φ.(cid:1)Finally the thesis follows from (3.1)
by recalling that (see, e.g. Lamperti (1977)) if φ ∈ D(B) then T φ ∈ D(B) and
t
BT φ= T Bφ, for each t ≥ 0. (cid:3)
t t
3.1. Dimensional reduction. We remark that the number of variable can be re-
d
duced by setting x = 1 − x . The new variable, that we still denote by x =
0 k
k=1
(x1...xd), lives in the closedPset
d
Σ = {x = (x ...x ) ∈ Rd : x ≥ 0 as k = 1...d, x ≤ 1}.
1 d k k
k=1
X
In all the following, we continue to write a and f for the respective functions de-
pending on the d variables x = (x ...x ) ∈ Σ. As or the non-local term J, it
1 d
8 A.L.AMADORI,A.CALZOLARI,R.NATALINI,B.TORTI
becomes
Jφ(x) = λ f (x)[φ(x+γ x (e −e ))−φ(x)]
ij i ij i j i
i,j=1...d
X
i6=j
+ λ f (x)[φ(x−γ x e )−φ(x)]
i0 i i0 i j
i=1...d
X
+ λ f (x) φ x+γ (1− x )e −φ(x) ,
0j 0 0j i j
" ! #
j=1...d i=1...d
X X
for any continuous scalar function φ∈ C(Σ;R).
Next subsections are devoted to the analytical study of the decoupled system
∂ u (x,t)+a(x)·∇u (x,t) = Ju (x,t), x ∈ Σ, t > 0
(3.3) t k k k
u (x,0) = x , x ∈ Σ, t = 0,
k k
(cid:26)
as k = 1...d.
3.2. Mild solutions. We establish global well-posedness of the Cauchy problem
(3.3). As a preliminary, we notice that the vector field a(x) does not point outward
at the boundary of Σ.
Remark 3.1. The projection of the vector field a orthogonal to the sides of the
boundary of Σ is always zero. Actually at x = 0 we have a(x)·e = a (x) = 0, while
k k k
d
at x = 1 we have
k
k=1
P
d d d d
a(x)· e = a (x) = − x f (x)+f¯(x) x = −xf(x)+f¯(x) = 0,
k k k k k
!
k=1 k=1 k=1 k=1
X X X X
d
because x = 1− x = 0.
0 k
k=1
P
As a consequence, the flux of the Cauchy problem for the autonomous equation
y˙ = a(y),
(3.4)
y(0) = x,
(cid:26)
is well defined and maps Σ into itself. It is worst noticing that it is nothing than
the solution to the replicator equation (1.1), after the dimensional reduction. In the
following, we shall write Y(x,t) for the solution of (3.4) starting at x. For any x,t,
thefunctions 7→ Y(x,s−t)isthecharacteristic lineof throughx,tfor problem(3.3).
Definition 3.1. A mild solution is a function u ∈C([0,T)×Σ;Rd) satisfying the
integral formula
t
(3.5) u (x,t) = Y (x,−t)+ Ju (Y(x,s−t),s)ds,
k k k
Z0
as k = 1...d.
Following the method of characteristics, we define
v(x,s,t) = u(Y(x,s−t),s)
RARE MUTATIONS IN EVOLUTIONARY DYNAMICS 9
and notice that for any t, v solves an integro-differential problem
∂ v (x,s,t) = Iv (x,s,t) x ∈ Σ, s > 0,
(3.6) s k k
v (x,0,t) = Y (x,−t) x ∈ Σ, s = 0,
k k
(cid:26)
as k = 1...d. Here
Iv (x,s,t) = λ f (Y(x,s−t))[v (x+δ (x,s−t),s,t)−v (x,s,t)]
k ij i k ij k
i,j=1...d
X
i6=j
+ λ f (Y(x,s−t))[v (x+δ (x,s−t),s,t)−v (x,s,t)]
i0 i k i0 k
i=1...d
X
+ λ f (Y(x,s−t))[v (x+δ (x,s−t),s,t)−v (x,s,t)],
0j j k 0j k
j=1...d
X
δ (x,r) =Y (Y(x,r)+γ Y (x,r)(e −e ), −r)−x,
ij ij i j i
δ (x,r) =Y (Y(x,r)−γ Y (x,r)e ), −r)−x,
i0 i0 i i
δ (x,r) =Y Y(x,r)+γ (1− Y (x,r))e , −r −x.
j0 0j i j
i=1...d
(cid:0) X (cid:1)
The Cauchy problem (3.6) is seen in the mild sense, actually
s
(3.7) v (x,s,t) =Y (x,−t)+ Iv (x,r,t)dr.
k k k
Z0
We next use a fixed point argument to solve (3.6). This brings a mild solution also
to (3.3) because
v (x,t,t) = u (x,t).
k k
The following proof is standard, but we report it for the sake of completeness.
Proposition3.2. Theproblem (3.3)hasanuniqueglobalsolutionu∈ C(Σ×[0,+∞);Rd).
Proof. Let S,T > 0, and χ the set of continuous scalar functions of (x,s,t) ∈ Σ×
[0,S]×[0,T], which is a Banach space endowed with the sup-norm. For k = 1...d
and r > 0 , let Bk be the set of functions of χ such that
r
sup{kv(x,s,t)−Y (x,−t)k : x ∈ Σ,0 ≤ s ≤ S,0 ≤ t ≤ T}≤ r,
k
and T the operator
s
Tv(x,s,t) = Y(x,−t)+ Iv(x,r,t)dr.
Z0
T maps B into itself because for v ∈ B we have
r r
kTv−v k≤ 2kfk skvk ≤ 2kfk S(r+kv k) ≤ r,
o ∞ ∞ o
provided that S ≤ r/2kfk (r+kv k). Moreover T is a contraction since
∞ o
r
kTv−Twk ≤ 2kfk skv−wk ≤ 2kfk Skv−wk ≤ kv−wk.
∞ ∞
r+kv k
o
It follows by contraction Theorem that (3.6) has an unique mild solution v ∈
C([0,S]×[0,S]×Σ) at least for S = r/2kfk (r +kv k). Moreover, since v ∈ B ,
∞ o r
we have kvk ≤ r + kv k. Then the fixed point argument can be iterated to get,
o
n
at any step n, a solution defined until S = 1 r + kr+kvok . As
n 2kfk∞ r+kvok (k+1)r+kvok
Sn → +∞, existence and uniqueness of a global solu(cid:18)tion followskP.=1 (cid:19) (cid:3)
10 A.L.AMADORI,A.CALZOLARI,R.NATALINI,B.TORTI
3.3. Viscosity solution. Using the tools of viscosity theory, one can eventually
prove the following result
Proposition 3.3. Let u and v be, respectively, viscosity sub/supersolutions of (3.3)
with u(x,0) ≤ v(x,0) for all x ∈ Σ. Then u(x,t) ≤ v(x,t) for all x ∈ Σ and t > 0.
Proof. The proof consists in assuming that there exists T > 0 so that
M = max{e−t(u(x,t)−v(x,t)) : 0 ≤ x ≤ 1, 0 ≤ t ≤ T}> 0
and getting a contradiction. Standard arguments exclude that any maximum point
x¯,t¯may have t¯=0 either t¯∈(0,T] and x¯ in the interior of Σ. We show that neither
x¯ in the boundary of Σ is allowed. To fix idea, suppose that x¯ is a strict maximum
d
point with x¯ = 0, x¯ > 0 as i= 2,...d and x¯ = 1. Next, a barrier function
1 i i
i=1
P
d
κ(x,t) = 1/x +1/(1− x )+1/(T −t)
1 i
i=1
X
is introduced and M is approximated by M(δ), the maximum value of
e−t(u(x,t)−v(x,t))−δκ(x,t).
It is clear that there exists a maximum point x(δ),t(δ) with x(δ) in the interior of Σ
and 0≤ t(δ) < T. Moreover
M(δ) → M > 0, x(δ) → x¯, t(δ) → t¯> 0, δκ(x(δ),t(δ)) → 0
as δ → 0. Now the perturbed functions u (x,t) = e−tu(x,t)−δκ(x,t) and v (x,t) =
δ δ
e−tv(x,t) can be handled by the standard tool of doubling variables and using equa-
tion (3.3). The step of passing to the limit as δ → 0 can be performed since
d
a (x(δ))
i
a (x(δ))
1 i=1
|δa(x(δ))·Dxκ(x(δ),t(δ))| ≤ x (δ) +(cid:12)(cid:12) dP (cid:12)(cid:12)δκ(x(δ),t(δ)),
(cid:12) 1 (cid:12) (cid:12) (x¯ −x (δ))(cid:12)
(cid:12) (cid:12) (cid:12) i i (cid:12)
(cid:12) (cid:12) (cid:12)i=1 (cid:12)
(cid:12) (cid:12) (cid:12)P (cid:12)
d (cid:12) d (cid:12)
(cid:12) (cid:12)
where the quantities |a (x(δ))/x (δ)| and | a (x(δ))/ (x¯ −x (δ))| are bounded
1 1 i i i
i=1 i=1
Pd P
because a is Lipschitz continuous and a (x¯), a (x¯) are zero by Remark 3.1. (cid:3)
1 i
i=1
P
In our particular setting, we have decided to emphasize the transport component
by following the method of characteristics. Nevertheless, the solution defined and
produced in the previous section coincides with the viscosity solution.
Proposition 3.4. Mild and viscosity solution of (3.3) coincide.
Proof. As well posedness holds in both framework, it suffices to check that the mild
solution is a solution in viscosity sense. We only prove the subsolution part, because
thesupersolutionis identical. Letk = 1...d, and φbea smoothscalar function such
that u −φ has a global maximum at (x,t) ∈ Σ×(0,+∞), with u (x,t) = φ(x,t).
k k
First, we notice that Ju (x,t) ≤ Jφ(x,t). Next, we set ψ(x,s,t) =φ(Y(x,s−t),s):
k